Transversals of squares

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Transversals in Latin Squares

A latin square of order n is an n×n array of n symbols in which each symbol occurs exactly once in each row and column. A transversal of such a square is a set of n entries such that no two entries share the same row, column or symbol. Transversals are closely related to the notions of complete mappings and orthomorphisms in (quasi)groups, and are fundamental to the concept of mutually orthogon...

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Transversals of Additive Latin Squares

Let A = {a1, . . . , ak} and B = {b1, . . . , bk} be two subsets of an Abelian group G, k ≤ |G|. Snevily conjectured that, when G is of odd order, there is a permutation π ∈ Sk such that the sums ai+bπ(i), 1 ≤ i ≤ k, are pairwise different. Alon showed that the conjecture is true for groups of prime order, even when A is a sequence of k < |G| elements, i.e., by allowing repeated elements in A. ...

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Transversals in generalized Latin squares

We are seeking a sufficient condition that forces a transversal in a generalized Latin square. A generalized Latin square of order n is equivalent to a proper edge-coloring of Kn,n. A transversal corresponds to a multicolored perfect matching. Akbari and Alipour defined l(n) as the least integer such that every properly edge-colored Kn,n, which contains at least l(n) different colors, admits a ...

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Latin Squares with No Transversals

A k-plex in a latin square of order n is a selection of kn entries that includes k representatives from each row and column and k occurrences of each symbol. A 1-plex is also known as a transversal. It is well known that if n is even then Bn, the addition table for the integers modulo n, possesses no transversals. We show that there are a great many latin squares that are similar to Bn and have...

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Latin Squares: Transversals and counting of Latin squares

Author: Jenny Zhang First, let’s preview what mutually orthogonal Latin squares are. Two Latin squares L1 = [aij ] and L2 = [bij ] on symbols {1, 2, ...n}, are said to be orthogonal if every ordered pair of symbols occurs exactly once among the n2 pairs (aij , bij), 1 ≤ i ≤ n, 1 ≤ j ≤ n. Now, let me introduce a related concept which is called transversal. A transversal of a Latin square is a se...

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ژورنال

عنوان ژورنال: Bulletin of the Australian Mathematical Society

سال: 1972

ISSN: 0004-9727,1755-1633

DOI: 10.1017/s0004972700045196